What is the average value of $8-2x^3$ on the interval $[2,4]$ ?
Solution: In general, this is the average value of function $f$ over the interval $[a,b]$ : $\dfrac{\int_a^b f(x)\,dx}{b-a}$ In our case, ${f(x)=8-2x^3}$, ${a=2}$ and ${b=4}$ : $\begin{aligned} \dfrac{\int_{ a}^{ b} {f(x)}\,dx}{ b- a}&=\dfrac{\int_{{2}}^{ {4}} ({8-2x^3})\,dx}{{4}-{2}} \\\\ &=\dfrac{\left[8x-\dfrac{x^4}{2}\right]_{2}^{4}}{2} \\\\ &=\dfrac{-96-8}{2} \\\\ &=-52 \end{aligned}$ In conclusion, the average value of $8-2x^3$ on the interval $[2,4]$ is $-52$.